I can remove a row from matrix
$M=\left(
\begin{array}{cccc}
m_{11} & m_{12} & m_{13} & m_{14} \\
m_{21} & m_{22} & m_{23} & m_{24} \\
m_{31} & m_{32} & m_{33} & m_{34} \\
m_{41} & m_{42} & m_{43} & m_{44}
\end{array}
\right)$
with factor like
$X=\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{array}
\right)$
by multiplying by it from left. The $X$ removes 3rd row:
$X M = \left(
\begin{array}{cccc}
m_{11} & m_{12} & m_{13} & m_{14} \\
m_{21} & m_{22} & m_{23} & m_{24} \\
m_{41} & m_{42} & m_{43} & m_{44}
\end{array}
\right)$
I can also remove column by
$Y=\left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0 \\
0 & 0 & 1
\end{array}
\right)$
by multiplying from right
$M Y = \left(
\begin{array}{ccc}
m_{11} & m_{12} & m_{14} \\
m_{21} & m_{22} & m_{24} \\
m_{31} & m_{32} & m_{34} \\
m_{41} & m_{42} & m_{44}
\end{array}
\right)$
Can I remove column by multiplying from left and remove row by multiplying from right?
Best Answer
For $A \in M_{p\times q}\left(\Bbb K\right)$ and $B \in M_{q\times r}\left(\Bbb K\right)$, $AB \in M_{p\times r}\left(\Bbb K\right)$$.
Your matrix is either $A$ or $B$ and the other one of those is the matrix you use to "remove" a line / column. So either $p$ and $q$ are fixed (meaning you modify your matrix by multiplying it by something on the right) and then you can only change $r$, the number of columns, or it's the other twi that are fixed so you can only change $p$, the number of columns.
So no, you can't.